A FORMULA TO CALCULATE THE SPECTRAL RADIUS OF A COMPACT LINEAR OPERATOR

,ABSTRACT. There is a formula (Gelfand’s formula) to find the spectral radius of a linear operator defined on a Banach space. That formula does not apply even in normed spaces which are not complete. In this paper we show a formula to find the spectral radius of any linear and compact operator T defined on a complete topological vector space, locally convex. We also show an easy way to find a non-trivial T-invariant closed subspace in terms ofMinkowski functional.


INTRODUCTION
In all that follows E stands for a linear space of infinite dimension over the field C of the complex numbers.E[t] will denote a complete topological vector space, locally convex, with topology and T:E --E will be a linear map.Finally, v(t) will be the filter of all balanced, convex and closed t- neighborhoods of zero (in E).
DEFINITION 1.The linear operator T E[t] E[t] is said to be a bounded (compact) operator, if there is a neighborhood U E v(t) such that T(U) is a bounded (relatively compact) set REMARK 2. It is easy to show that any compact operator is a bounded operator and any bounded operator is continuous.DEFINITION 3.For a topological vector space X[O] and a linear operator S :X[0] X[O] we define the resolvent of S as po(S) { C[I-S: X[O] -X[O] is bijective and has a continuous inverse}.
The spectrum of S is defined by ao(S) C\pt(S) (the set theoretic complement) and the spectral radius DEFINITION 4. A net {x}2 C E[t] is said to be t-ultimately bounded (t-ub) if given any V v(t) there is a positive real number r and an index a0 J, both depending on V, such that x E rV V a >_ ao.Let us denote by F the set of all t-ub nets in E [t].586 F G BONALES AND R V. MENDOZA REMARK 5. Any bounded net is a t-ub net.For more details about t-ub nets we refer the reader to DeVito [2] From now on T :E[t] --} E[t] will denote a compact operator and U E v(t) will be the zero- neighborhood such that T(U) is a t-relatively compact set.Pv will stand for the Minkowski functional generated by U (see Coflar ]), which is a seminorm on E. Let us denote by E[Ptr] the linear space E with the topology given by the seminorm DEFINITION 6 (r'-convergence).Let E C, with : 0 We will say that T -0 (T T o T o o T n times) if given both V v(t) and {xo).F there exist a0 J and no N such that T (x,) E V V a E a0 and V n > no.7 0}.
REMARK $.It is shown by Vera [7] that for a bounded operator T, (a) 7t (T) < oo, and for any C such that 7 (T) < ][, T" _r.k 0 co) srt(T) < 7t(T), where 8rt(T) is the spectral radius of T. (c) If E[t] is a Banach space then 7t (T) srt (T)   The main theorem (Theorem 28) in this paper states that for a compact linear operator on any complete topological vector space, locally convex; we have srt (T) 7t (T), even when the topology t is not given by a norm.In fact, Remark 8 (c) will be used to prove our result.

MAIN RESULTS
Now, we will state a well known theorem about compact operators THEOREM 9 (see Nikol'skij [5]).The spectrum of a compact operator T on an infinite- dimensional linear topological space E consists of zero and no more than a countable set of eigenvalues different from zero.The unique accumulation point ofthis set, if it is infinite, is zero.
REMARK 10.The topology on E given by the seminorm Ptr is coarser than the topology ( _< ).
PROPOSITION 11.T E[Pu] E[Pu] is a compact operator.
PROOF.Since T(U) is t-relatively compact and Pu <_ , T(U) is also Pu-relatively compact.
l}t'mrnoN 12. -r -ire Ill T 0 Hrw, the meaning of T 0 is given by Definition 6 where the topology Ptr is used instead of It is easy to show that F ev-convergence means that given any net {zo}j C E such that for all a J, Pv(zo)5 r for some r R + (these kinds of nets are said to be Ptr-bounded nets), then Pt( T"zo) --}0 as a net in R whose set ofindices is N x J.
PROOF.Let   C be such that 7(T) < [[, let V E v() and {zo)j F be given Since T(U) c V DeVito [2] shows !T(U) is a bounded set, there is a positive real number that {zo) F = {rl:o).rF. This implies that there exist both c0 J and r > 0 such that rlZo r2UVc > co.This means that Ptr(rzo) A r2, hence the net {zo)>,0 is a Ptr-bounded net.
15. Since U is a bc, convex d closed set, for y re 0 S r < 1, {x E lPv(x) r} C U. Then N C U. Since is not ven by a no it follows om a Theorem of Kolmogoroff (Theorem 1.39 in Rudin [6]) that {0} N. On the other hd, if T 0 then N E.
The follong theorem is a generition of a theorem of Lomonosov [4] about non-tfi invt subspas.
OM 16.N is a clo lin mbspace of E d T(x) 0 for l x N. In picd, N is a T-vt mbspa.
Deflation 14 tes us that v is well defined.It is to show that v is a no in E/N We 11 denote by (E/N)[v] the veor space E/N th e topolo ven by the no .
" E/N E/N be defined by (x + N) T(x) + N.
By Theorem 16 T is a well defin map.
PROSON 22.   C{0} is eigenvue of T if d oy if it is eigenvue of T. PROOF.L 0 C be eigenvue of T. Henceereests x+NcN such that (x+N)=(x+N)T(x)+N=(x)+Nx-T(x)ENyN such that (I-T)(x) y.By Theorem 16 T(y) 0, hence T(x y) T(x) (x y) where x y 0 because z N. Therefore is eigenvue of T.
C{0} is eigenvue of ifd oy if it is eigenvue of.588 F G BONALES AND R.V MENIX)ZA PROOF.Let q E/N, q 0 be such that (q) q Let (z, -b !V} C E/IV be such that it converges to q Hence {z, + N) is a Prj-bounded set and therefore, because is a compact operator, "(z, + At)) has a convergent subsequence.Without loss of generality let us suppose that (z.+ N) ---, z + N E/N, then q (q) z + N which implies that q (z + N) E/N, hence is an eigenvalue of The proof of the second part ofthis theorem is trivial since PROPOSITION 26.
(9) < % ()   The Immna follows from the definitions of-7 v (') and "7u () and the fact that is the restriction of" to E/N.